Solving 10^b = 91 To The Nearest Ten-Thousandth

Introduction: Unveiling Exponential Equations

In the realm of mathematics, exponential equations hold a unique allure. These equations, characterized by a variable nestled within the exponent, often appear daunting at first glance. However, with the right tools and techniques, we can unravel their mysteries and arrive at elegant solutions. This article delves into the intricacies of solving the exponential equation $10^b = 91$, guiding you through the process of finding the value of b to the nearest ten-thousandth. We will explore the fundamental concepts underlying exponential equations, the power of logarithms in simplifying them, and the practical application of these principles to arrive at our desired solution. Understanding exponential equations is crucial as they appear in various scientific and engineering applications, including population growth, radioactive decay, compound interest calculations, and more. By mastering the techniques to solve these equations, you will enhance your mathematical toolkit and gain a deeper appreciation for the interconnectedness of mathematical concepts.

At the heart of the equation $10^b = 91$ lies the concept of exponential growth. The base, 10 in this case, is raised to the power of the exponent, b, resulting in the value 91. Our mission is to determine the specific value of b that satisfies this relationship. A naive approach might involve trial and error, guessing values for b and evaluating $10^b$. However, this method is inefficient and unlikely to yield a precise answer. Instead, we will employ a powerful mathematical tool: logarithms. Logarithms provide a systematic way to "undo" exponentiation, allowing us to isolate the variable b. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. In our case, we will use the common logarithm (base 10), denoted as log₁₀ or simply log. Applying the logarithm to both sides of the equation will transform it into a more manageable form, paving the way for a straightforward solution.

Understanding Logarithms: The Key to Unlocking Exponents

To effectively tackle the equation $10^b = 91$, a solid understanding of logarithms is indispensable. Logarithms are the inverse operation of exponentiation, providing a way to isolate exponents within equations. The expression logₐ(x) = y is equivalent to aʸ = x, where a is the base, x is the argument, and y is the logarithm. In simpler terms, the logarithm (y) is the power to which the base (a) must be raised to obtain the argument (x). When the base is 10, we have the common logarithm, denoted as log₁₀(x) or simply log(x). This implies that log(x) = y if and only if 10ʸ = x. For example, log(100) = 2 because 10² = 100, and log(1000) = 3 because 10³ = 1000. Understanding this fundamental relationship between logarithms and exponents is crucial for solving exponential equations.

The properties of logarithms are particularly useful when dealing with complex expressions. One of the most important properties is the power rule, which states that logₐ(xⁿ) = n * logₐ(x). This rule allows us to bring exponents down as coefficients, simplifying equations significantly. In the context of our problem, $10^b = 91$, applying the logarithm to both sides and utilizing the power rule will enable us to isolate b. Another essential property is the logarithm of the base itself: logₐ(a) = 1. For the common logarithm, this means log₁₀(10) = 1, which will be instrumental in solving for b. The ability to manipulate logarithmic expressions using these properties is a cornerstone of solving exponential equations efficiently and accurately. Furthermore, understanding the relationship between exponential and logarithmic functions provides a deeper insight into mathematical concepts, allowing for problem-solving in various scientific and engineering contexts. For example, these concepts are pivotal in understanding exponential growth and decay models, which are widely used in fields like finance, biology, and physics.

Solving the Equation: A Step-by-Step Approach

Now, let's embark on the journey of solving the equation $10^b = 91$ using the principles we've discussed. The initial step involves applying the common logarithm (base 10) to both sides of the equation. This gives us log(10^b) = log(91). The reason for this step is to leverage the properties of logarithms to isolate the variable b. Applying the logarithm to both sides maintains the equality of the equation, setting the stage for further simplification.

Next, we invoke the power rule of logarithms, which states that logₐ(xⁿ) = n * logₐ(x). Applying this rule to the left side of our equation, log(10^b), yields b * log(10) = log(91). This crucial step brings the exponent b down as a coefficient, making it easier to solve for. Now, we recall the fundamental property of logarithms: logₐ(a) = 1. In our case, this translates to log₁₀(10) = 1. Substituting this into our equation, we have b * 1 = log(91), which simplifies to b = log(91). At this point, the problem reduces to finding the value of log(91), which can be obtained using a calculator. The value of log(91) is approximately 1.959041475. However, we need to round the result to the nearest ten-thousandth, which is four decimal places. Therefore, the solution to the nearest ten-thousandth is b ≈ 1.9590.

Verifying the Solution and Exploring Alternatives

To ensure the accuracy of our solution, it's essential to verify it. We found that b ≈ 1.9590, so we can substitute this value back into the original equation: $10^b = 91$. Evaluating $10^{1.9590}$ using a calculator, we get approximately 90.9988, which is very close to 91. The slight discrepancy is due to the rounding we performed to the nearest ten-thousandth. This verification step confirms that our solution is indeed accurate within the specified precision.

While we successfully solved the equation using common logarithms (base 10), it's worth noting that we could have used logarithms with any valid base. For instance, we could have used the natural logarithm (base e), denoted as ln(x). Applying the natural logarithm to both sides of the equation $10^b = 91$ would give us ln(10^b) = ln(91). Using the power rule, we get b * ln(10) = ln(91). To solve for b, we would then divide both sides by ln(10), resulting in b = ln(91) / ln(10). Evaluating this expression using a calculator, we find that b ≈ 1.9590, which is the same solution we obtained using common logarithms. This demonstrates the flexibility of logarithms and reinforces the idea that the choice of base does not affect the final solution, as long as the logarithmic properties are applied correctly. Exploring alternative approaches not only solidifies our understanding but also expands our problem-solving toolkit, allowing us to tackle similar problems with greater confidence.

Conclusion: Mastering Exponential Equations

In this article, we embarked on a journey to solve the exponential equation $10^b = 91$ to the nearest ten-thousandth. We began by understanding the fundamental concepts of exponential equations and the crucial role of logarithms in simplifying them. We explored the properties of logarithms, particularly the power rule, which enabled us to isolate the variable b. By applying the common logarithm to both sides of the equation and leveraging the power rule, we transformed the equation into a manageable form. We then solved for b and obtained a value of approximately 1.9590.

To ensure the accuracy of our solution, we performed a verification step by substituting the value of b back into the original equation. The result confirmed that our solution was accurate to the nearest ten-thousandth. Furthermore, we explored an alternative approach using natural logarithms, demonstrating the flexibility of logarithmic principles and reinforcing the concept that the choice of base does not impact the final solution. Mastering the techniques for solving exponential equations is invaluable in various fields, including science, engineering, and finance. These equations appear frequently in models describing growth, decay, and other phenomena. By understanding the properties of logarithms and practicing their application, you can confidently tackle a wide range of mathematical challenges. The solution to the equation $10^b = 91$ to the nearest ten-thousandth is B. 1.9590.

This exploration not only provides the answer to a specific problem but also equips you with the tools and understanding to approach similar problems with confidence. Continue practicing and exploring the fascinating world of mathematics, and you will undoubtedly unlock even greater insights and problem-solving abilities.